6. assignment problems

ADVANCED OPERATIONS

RESEARCH

By: -Hakeem–Ur–Rehman

IQTM–PU

A ROASSIGNMENT PROBLEMS

ASSIGNMENT PROBLEMS The assignment problem refers to the class of LP problems

that involve determining the most efficient assignment ofresources to tasks

The objective is most often to minimize total costs or totaltime to perform the tasks at hand

One important characteristic of assignment problems isthat only one job or worker can be assigned to onemachine or project

An assignment problem can be viewed as a transportationproblem in which the capacity from each source is 1 andthe demand at each destination is 1

3

Let m be the number of jobs as well as the operators, and tij be the processing time of the job i if it is assigned to the operator j. Here the objective is to assign the jobs to the operators such that the total processing time is minimized.

Operators

Job

1 2 … j … m

1 t11 t12 t1j t1m

2

.

i ti1 tij tim

.

m tm1 tm2 tmj tmm

GENERAL FORMAT OF ASSIGNMENT PROBLEM

4

Examples of assignment problem

Row entity Column entity Cell entity

jobs operators Processing time

Programmer program Processing time

operators machine Processing time

Drivers Routes Travel time

Teachers Subjects Students pass

percentage

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Assignment problem as a zero-one ( Binary) programming problem

Min Z= c11x11++cijXij+.+cmmXmm=

Subject to x11+…………...+x1m =1x21+…………...+x2m =1……..

xm1+…………...+xmm =1x11+…………...+xm1 =1x12+…………...+xm2 =1………………..

x1m+…………...+xmm =1xij.=0 or 1 for i=1,2….m and j=1,2…..m.

THE HUNGARIAN METHOD (FLOOD’S TECHNIQUE)

Test opportunity cost table to seeif optimal assignments arepossible by drawing the minimumpossible lines on columns and/orrows such that all zeros arecovered

Find opportunity cost

(a) Subtract smallest number ineach row from every number inthat row, then

(b) subtract smallest number ineach column from everynumber in that column

Set up cost table for problem Revise opportunity cost table intwo steps:

(a) Subtract the smallestnumber not covered by a linefrom itself and every otheruncovered number

(b) add this number at everyintersection of any two lines

Optimal solution at zerolocations. Systematically makefinal assignments.

(a) Check each row and columnfor a unique zero and make thefirst assignment in that row orcolumn

(b) Eliminate that row andcolumn and search for anotherunique zero. Make thatassignment and proceed in a likemanner.

Step 1

Notoptimal

Step 2

Optimal

SPECIAL CASES IN ASSIGNMENT PROBLEMS

Unbalanced Problem

Maximum Case

Multiple Alternative Solution

Prohibited Assignment

ASSIGNMENT PROBLEMS USING EXCEL SOLVER

The Fix-It Shop has three rush projects to repair. They have three repairpersons with different talents and abilities. The owner has estimates ofwage costs for each worker for each project. The owner’s objective is toassign the three project to the workers in a way that will result in thelowest cost to the shop. Each project will be assigned exclusively to oneworker.Estimated project repair costs for the Fix-It shop assignment problem

PROJECT

PERSON 1 2 3

Adams $11 $14 $6

Brown 8 10 11

Cooper 9 12 7

ASSIGNMENT PROBLEMS USING EXCEL SOLVER

ASSIGNMENT PROBLEMS USING EXCEL SOLVER

QUESTIONS